Abstract
This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type.
Highlights
In [1], Hardy established that ∑ ∑ ∞ ∞ j=1 i=1 φiψj i+j ≤ π sin π α ∞ ∑ φiα i=1 1 α ∑ β ψj j=1 1 β (1)
The organization of the paper is as follows: in Section 2, we show some basics of the time scale theory and some lemmas on time scales needed in Section 3 where we prove our results
We proved the reverse Hilbert-type inequalities on time scales which involve nonnegative, concave, and supermultiplicative functions
Summary
Some authors established the reverse Hölder inequalities, the reverse Young inequalities, and the reverse Hilbert inequalities by using the Specht’s ratio function, see [8,9,10,11,12]. Zhao and Cheung [11] established the reverse Hölder inequalities by using the Specht’s ratio function and proved that if ψ(ζ) and (ζ) are nonnegative continuous functions and ψ1/α(ζ) 1/β(ζ) is integrable on [a, b], ψα(ζ)dζ β(ζ)dζ ≤ In [12], Zhao and Cheung established the reverse Hilbert inequalities by using the Specht’s ratio and proved that if 0 ≤ α, β ≤ 1, {λi}, {ψj} are nonnegative and decreasing sequences of real numbers for i = 1, 2, ..., k and j = 1, 2, ..., r with k, r ∈ N, ∑ ∑ k r Sα,β,k,r,i,j ∑is=1 λs α ∑tj=1 ψt β 1 i=1 j=1 (ij) 2
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